If you use the Invite Others... button in a message…
…I cannot invite a group to the message, even though I have the necessary permissions.
I currently have the same issue for topic invites, although I am sure they have been working at least sometimes.
Group PMS and mentions are very rough, we plan on revamping them in an upcoming release.
Just faced a similar issue, which could be connected to this post.
I can only invite one person to a PM at a time, after you fill-out the username, there’s no way you can add another one in the same invitation. Haven’t tried it with groups or Emails. I’m just trying to add multiple users to the same conversation at the same time.
Not being able to invite “groups” is definitely something I want fixed for this release.
At the moment if you want to invite multiple users you invite them one at a time, but agree the UI there can be improved, it kind of invites you to enter more users but does not expose a way to do it.
I just came here looking for this - it would be a big help to be able to invite discourse groups here, e.g. to pull all moderators into a private message.
Thanks! 
Oh, and while we’re at it - it’s also not possible to remove a group from a message. There is no X next to the group name.
Yes, I often get direct messages to me that should be handled by an existing group (of which I’m not a member!).
On our site we’re transitioning from a setup where automated messages were previously being sent to users and moderators mentioned individually by username. Now automated messages will be sent to the user and a group, e.g. the moderators group. We’ve been experimenting and there are some messages that now include incorrect groups and users.
So… to help clean this up I’d like to be able to update messages to reflect this but am finding it impossible to do so. 
The following new features appears to be needed for my use case, and I think would probably help everybody as we start to use messaging more:
When looking at individual messages, the ability to remove (x out) currently included groups and to add new groups.
The ability to bulk update messages to add/remove users and groups, at least via a query. I’d be afraid to try unless someone has done it successfully already and can share the query. 
A UI for this might be worthwhile too, e.g. by adding an action to group message lists.
Per:
https://github.com/discourse/discourse/commit/886616987983b6a9f532210e8598a9b5f0e60ed1
Groups can be invited and removed from private messages.
